Casimir effect in a weak gravitational field

F. SorgeUniversità di Padova and I.N.F.N., Italy

Padova, january 25, 2007

[Class. Quantum Grav. 22, 5109 (2005)]

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Motivations • To analyze the influence of gravity upon

Casimir effect, comparing present with other existing results.

• Possible relevance in some cosmological scenarios (dark energy?).

• Validity of General Relativity in the limit of very small distances.

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Casimir Effect(static)

Conducting plates

L

There is a small (attractive) force between the plates

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Casimir effect can be defined as the stress on the bounding surfaces (of a cavity) when a quantum field is confined in a finite volume of space. The boundaries may be real material media, or the interface between two different phases of the vacuum of a given field theory.

The boundaries may also represent the non-trivial topology of space. In any case field confinement restricts the modes of the quantum fields giving rise to sometimes important and measurable macroscopic effects.

Although commonly considered a somewhat exotericeffect, the Casimir effect could play a relevant role in various contexts, and at various length scales - from particle confinement (as in some interquark models) to the large scale structure of the Universe.

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References• [1] Casimir H 1948 Proc. K. Ned. Akad. Wet. 51 793• [2] Casimir H and Polder D 1948 Phys. Rev. 73 360• [3] Milton K A 2001 The Casimir Effect: Physical Manifestations of

Zero-Point Energy (River Edge: World Scientific) • [4] Bressi G, Carugno G, Onofrio R, and Ruoso G 2002 Phys. Rev.

Lett. 88 041804• [5] Bimonte A, Calloni E, di Fiore L, Esposito G, Milano L and Rosa

L 2003 Preprint hep-th/0302082 v1• [6] Bimonte G, Calloni E, Esposito G, Rosa L (2006), Contributed to

17th SIGRAV Conference• [7] Bimonte G, Calloni E, Esposito G, Rosa L (2006),

Phys.Rev.D74:085011. • [8] Caldwell R R 2002 Preprint astro-ph/0209312 v1• [9] Unruh W G 1976 Phys. Rev. D 14 870• [10] Sorge F (2006) Int. J. Mod. Phys. A 21, 6173

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Weak field approximation

Space-time around a rotating object in terms of gravito-electromagnetic potentials Ai and Φ:

(Lorentz gauge)

(metric)

where:

We assume a non-rotating source : Ai = 0.

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The resulting metric is diagonal:

Although the metric recalls the flat minkowskian space-time case, we must bear in mind that the rectangular coordinates here employed are not the Minkowskian ones.

The simple form of the metric has been obtained imposing the Lorentz gauge; this, in turn, obscured the naive physical meaning of the coordinates.

Dutiful care is needed in extracting physical information, also in this weak field approximation.

Notice:

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Quantum field in a rigid cavity

Gravitational source

Φ - potential expansion:

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For example:

Notice that x- and y- dependence of Φ yields o(M/R)3

contribution only, due to the small cavity size.

The metric reads now:

both constants

Also:

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The metric determinant is:

We introduce the following static observer:

All the measurements are referred to such observer.

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Basic tools for QFCS

K-G eq. for a scalar massless field

In vacuum and with presents assumptions and approximations, K-G reads:

Define a scalar product as: spacelike surface

Determinant of the inducedmetric on Σ

Time-like unit vectorothogonal to Σ

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Natural choice:

Energy-momentum tensor:

Mean vacuum energy density inside the cavity (Casimirenergy density):

(proper value)Proper cavity volume

Proper vacuumenergy density

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Casimir energy:the zero-order approximation

Flat space-time case:

(Dirichlet boundary

conditions)

(Eigenmode frequencies)

Zero-order approximation

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The relevant bilinear component Ttt reads:

Also: so that:

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proper-time Schwinger representation

Infinite sum: Riemann ζ - function regul.

… the expected Casimir effect result in flat space-time.

Performing…

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Casimir energy:the first-order approximation

Gravitational field up to o(M/R) i.e., neglecting the o(M/R)2 γ – terms:

(K.-G. eq.)

Field positive-energy modes satisfying Dirichlet b.c.’s:

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From K.-G. equation:

first-order approximation

From the above defined scalar product:

the normalization constant follows:

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Proceeding as above, we find the vacuum energy densityneglecting the o(M/R)2 terms. So we set:

The relevant bilinear component Ttt reads now:

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Vacuum energy density:

Using the above found eigenfrequencies we rewrite:

Performing: and regularizing, we find:

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But this is NOT the whole story…

We need to express L in termsof the proper distance Lp :

…just the usual flat space-time Casimir result!

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The lesson…… we learned from the first-order approximation is that Casimirenergy is unaffected by a Φ = const gravitational field. This is notsurprising: at the first order of approximation Φ acts as a uniform, static background gravitational potential. So, no measurable physicaleffects are expected, as such a potential can be gauged away (at anyorder).

We conclude that - if gravity does affect Casimir energy - measurableeffects lie beyond the first order approximation, where Φ begins toshow its spatial dependence.

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Casimir energy:the second-order approximation

We now move to explore o(M/R)2 effects.

Thanks to the lesson of the above first-order approximation, weknow that the background static potential Φ = const can be discarded(as a uniform potential) at any order in M/R. It could play a role onlythrough terms as

or so, which are ruled out in the present o(M/R)2 – approximationscheme.

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Henceforth, we will set Φ0 = 0 in all the subsequent calculations.

Seek for solutions of K.-G. [to o(M/R)2] order such as:

(Airy equation)

(K-G.)

Defining:

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General solution in terms of Bessel functions:

constants

Notice:

So we employ the asymptotic form:

constants

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Dirichlet b.c.’s at the cavity plates:

(p, q integers)

expanding around the (small) parameter La/b we get:

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o(M/R)2

Vacuum energy density (full form):

Notice that the integrand does depend on z, due to the γcontributions.

From:

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A short-cut to avoid evaluating the cumbersome bilinearterm: is as follows:

Order-by-order expansion:

Accordingly:

Up to

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Some algebra yields:

Leading term

Flat space-time Casimir result

o(M/R)2

contribution

Using: (Φ0 = 0)

gives:

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Performing the remaining integration, summation and ζ -functionregularization as above, we immediately find the renormalizedvalue:

It seems just the flat space-time Casimir result!

But now we still need to change the coordinate cavity size L into the proper value Lp…

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Inverting and substituting we get:

And:

Finally, notice that the last term contributes with o(M/R)2

terms only; so the volume integration and the projection on the observer's frame can be carried out as in a flat space-time.

Clearly, the corresponding correction to is simply a manifestation of the direct interaction between the gravitationalfield and the quantum field inside the cavity.

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Since this result has to be ultimately evaluated for a staticobserver in a flat metric, it must coincide with the correspondingcorrection to the flat space-time Casimir result.

All that allows us to find out such second correction in a straightforward way.

Comparing …

we easily argue that the second o(M/R)2 correction to the flatspace-time Casimir energy is simply +γ L times the flat space-time case value, so:

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Notice: L has been replaced by the proper size Lp; indeed, thisis correct up to the o(M/R)2 order.

Putting all things together we finally have:

This is Casimir vacuum energy in presence of gravity, up to second order in (M/R).

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Gravity reduces the absolute value of (negative) Casimir energy:

the attractive forceobserved between the cavity plates must decrease:

Indeed, a very small correction!!!

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Conclusions

• Casimir effect is not influenced by a weak static spatially uniform gravitational field.

• Casimir energy is modified in presence of a spatially non-uniform gravitational field.

• The absolute value of ECas is slightly reduced by gravity.

• The corresponding attractive force between Casimir plates weakens.

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Final Remarks• If the Equivalence Principle holds, then a similar force

weakening should be observed also in an ideal laboratoryexperiment, where a Casimir cavity undergoes a (strong) acceleration phase [Sorge F (2006) Int. J. Mod. Phys. A 21, 6173].

• It is likely that, in a cosmological scenario, the interaction between vacuum energy and gravity can become relevantin determining the dynamics of the universe (dark energy?)

• It is still a puzzling, open question whether GeneralRelativity can really extend its realm to microscopicscales. In this respect, a deeper understanding of the physics involved in Casimir effect seems of fundamentalinterest, both from a theoretical and experimental point of view.

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Thank You

very much